In the area of constitutive modeling of mechanical behavior of engineering materials, finite deformation has become a subject of intense interest in the mechanics community. Design of equipment and processes for finite deformation is usually supported by calculations involving stress and deformation analysis of metal components at large plastic deformation level. These calculations include the problems of fabrication processes, metal forming, and impact.
Theorists have enjoyed certain degrees of success by proposing numerous concepts such as elastic-plastic decomposition of deformation gradient, co-rotational rate for the kinematical and state variables, plastic spin, choice of unstressed configuration, distinction between the kinematics of continuum and its underlying substructure, representation of substructure and texture, physical interpretation of internal variables, and the evolution of these variables. Computation and implementation of constitutive models at the finite strain range are available. However, the application of computer codes to predict plastic deformation responses has had limited success. While the computational schemes in modern codes have reached a high level of sophistication, these advances have not been matched with the development of more accurate constitutive equations.
In the development of more accurate constitutive equations, there are disparate views on the manner in which material deformation and rotation affect the evolution of anisotropy. Some of these disparities may be resolved by further theoretical investigation; however, some will have to rely on experiment, particularly under multiaxial loading condition. Additional investigation including strain hardening, yield surface measurements, cyclic loading, and strain-rate effect would provide an extensive material data base for use in the development of constitutive equations.
Most available experimental results for deformation of metals under multiaxial loading condition are for small strain range. This is largely due to the limitation of transducers used in the strain measurement. Typically, strain gages and commercially available axial-torsional extensometers (Instron and MTS) have a limit of less than two percent strain. Some experimental data for combined axial-torsional loading of metal tubes have been reported in this strain range. These data are extremely useful in the development and verification of constitutive models at small strain level.
Transducers that measure and control normal and shear strains at large strain range are not generally available, particularly for combined axial-torsional loading condition. A need exists for this kind of experimental data which may be used by the theorists to develop and verify constitutive models at large strain range. In addition, the experimental investigation has an independent value and its importance should be recognized in its own right.
The design of strain transducer for finite strain requires mastering of sophisticated techniques in engineering. Several technical problems need to be overcome. Two factors affect the strain range of an experiment -- the specimen geometry and test method, and the strain measuring device.
Different specimen geometries and test methods have been reported and are the primary factor in the determination of strain limits. Uniaxial tension or compression with specimen remachining can lead to large plastic strain. However, remachining interrupts the experiment which is undesirable. Also, lubrication is an important issue in compression tests. Torsion of very short thin-walled tubes, or the "Lindholm tube geometry", suffers from a second order length change and buckling instabilities. End effects may also influence stress uniformity. These two types of tests do provide valuable information concerning texture development at very large strain. Other tests include torsion of round solid bars which has strain gradient and indirect tests such as wire drawing and tension, etc. Although these tests are applicable for very large strain range, complications do arise in these tests.
Combined axial-torsion of thin-walled tube is an ideal test for the study of constitutive equation. This test is, however, limited to moderately large strain range due to buckling or fracture depending on material tested and specimen geometry. Even though it is known that in this moderate strain range, the change of texture is not significant, the combined axial-torsional test in this strain range is still a valuable test in that it may be used to investigate the anisotropic rate of strain hardening, cyclic properties of material, and fatigue life due to cumulative damage. The distortion of yield surface and the path dependent rate of kinematic hardening have occurred at small plastic strain range and their occurrence at moderately large strain range is of special interest. Furthermore, this test is suitable for the investigation of effects due to rotation, multi-axial stress/strain coupling, and stress/strain cycling.
A brief summary of existing strain transducers for axial-torsional test follows:
Extensometer of Brown and Miller: The extensometer is attached to the specimen by three tool-steel pins at each end of the 25 mm gage length. The pins are spring loaded to compensate for the radial strains arising in the specimen. The axial strain is measured by the average of two linear variable differential transformer (LVDT) readings to eliminate bending errors, and the torsional strain is determined from a wire-wound potentiometer whose spindle and body, respectively, are rotated by a pair of pulleys, one from each end of the gage length. The gage length is set initially by plastic spacing pieces to within plus or minus 0.3%. Applications to axial strain of up to 2% and shear strain of up to 4% have been reported.
Strain Measuring Device of Moon and Bell: The axial strain and the angle of twist are measured independently. This is accomplished by means of small brass collars glued to the specimen at a fixed distance apart. The collars can rotate in well-lubricated guide disks, which can slip up and down along well-lubricated guide rods. Two clip gages were attached to the opposite sides of these guide disks. Thus, any longitudinal elongation is transmitted to these clip gages by the guide disks. The angle of twist is measured by means of a wheel attached to the lower grip and a vernier scale. Strain measurements up to 12% axial and 27% torsional have been achieved. However, shear strain is obtained from the rotation of lower grip, not representative of the actual shear strain of the gage section; furthermore, brass collars restrict diametral change during experiment.
Extensometer of Khan and Parikh: This is an updated version of the design of Moon and Bell. The improvement is in the use of a rotary variable differential transducer (RVDT) to measure the angle of twist between the two grips. A rubber ring and a wheel are used to transmit the rotation of the grip to the RVDT. However, the two drawbacks cited in connection with the device of Moon and Bell still exist in this version.
Extensometer of Socie: The extensometer is inside the tubular specimen. This is a special design to allow for outside surface of the specimen easily replicated with acetyl cellulose film to monitor crack formation and growth. Axial displacements are measured and controlled with a LVDT located on the centerline of the specimen and rotations measured with a RVDT. Applications to axial strain of up to 1% and shear strain of up to 1.5% have been reported.
Extensometer of Liu: The strains are calculated from the relative displacements measured from two reference gage points assumed on the uniform section of the specimen. The basic elements are two assemblies of universal joints; each operates independently to monitor the motion of the respective reference gage point via a ceramic extension probe. Three cantilever-type transducers are used; two are for the circumferential displacement measurements and a third for the axial differential displacement. The device is capable of measuring plus or minus 5% axial and plus or minus 5% shear strain, with a gage length of 20 mm.
Capacitance Strain Transducer of Yeakley and Lindholm: The extensometer mounts inside the tubular specimen and grips. It is water cooled for elevated temperature operation. The extensometer consists of an armature (or rotor) containing one set of capacitance plates and a concentric housing (or stator) containing a second set of plates. The extensometer is supported on axes inside the hollow tubular specimen by six spring-loaded reference arms. The arms have knife-edge tips at the end where they contact the inner wall of the specimen. The gage length is normally 12.7 mm, but it can be changed to give a variable gage length of 1 to 6 cm. The armature contains a set of four active capacitance plates. Each covers a 45 degree arc. The armature plates are driven with a constant amplitude carrier signal and act as the driven elements for both axial motion and rotation motion. The housing contains ten separate capacitance plates. The two outer rings along with the armature form two arms of a capacitance half-bridge sensitive to axial strain only. The center segments are alternately connected to form the two output capacitance elements for rotational measurement. Along with the armature plates, these form two arms of a torsional-strain half-bridge. The axial dynamic range is 10% and torsion is plus or minus 22.5.degree.. The structure of this extensometer is complicated.
As discussed in the previous section, an axial-torsional extensometer for large strain range is still not available commercially. Although there are several devices designed by research workers and have appeared in the literature, they all have limitations in terms of suitable type of experiment, strain limits, whether or not the measured shear strains are truly representative of strains at the gage section of the specimen, and error bounds.
Those concerned with these and other problems recognize the need for an improved axial-torsional extensometer.